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Axiomatix(NASA-CR-167459) ENGINEERING EVALUATICNS Nd2-15114AND STUDIES. REFUhT FUR IUS STUDIES FinalAnnual Report, 1981 (Axicnatix, Los Angeles,Calif.) 44 p HC AQJ/dF A01 CSCL 22B Unclas
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ENGINEERING EVALUATIONS AND STUDIES
ANNUAL F M1 REPORT FOR IUS STUDIE OContract No. NAS 9-16067, Exhitit A
Prepared for
' Lyndon B. Johnson Space CentErHouston, Texas 7705E,
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Prepared by
y Axiomatix9841 Airport Blvd., Suite 912Los Angeles, California 90045
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i Axiomatix Report No. R8110-3October 20, 1981
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TABLE OF CONTENTS
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Page
1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Statement of Work . . . . . . . . . . . . . . . . . . 1
1.1.1 Objectives . . . . . . . . . . . . . . . 11.1.2 Stipulated Tasks . . . . . . . . . . . . . . . 1
2.0 GENERAL APPROACH . . . . . . . . . . . . . . . . . . . . . 2
3.0 TECHNICAL INVESTIGATION . . . . . . . . . . . . . . . . . 3
3.1 Analysis of IUS STDN/TDRS Transponder Performance . . 3
3.1.1 IUS Memo No. 125.
.
.
. . . . . . . . . . 93.1.2 IUS Memos No. 112and118 . . . . . . . . 103.1.3 IUS Memos No. 114 and 116 . . . . . . . . . . 113.1.4 IUS Memo No. 115 . . . . . . . . . . . . . . . 133.1.5 IUS Memo No. 117 . . . .. . . . . . . . . . 163.1.6 IUS Memo No. 123 . . . . . . . . . . . . . . . 183.1.7 IUS Memo No. 110 . . . . . . . . . . . . . . . 203.1.8 IUS Memo No. 124 . . . . . . . . . . . . . . . 213.1.9 IUS Memos No. 122 . . . . . . . . . . . . . . 243.1.10 IUS Memos No. 119 and 120 . . . . . . . . . 263.1.11 IUS Memo No. 121 . . . . . . . . . . . . . . . 28
4.0 CDR ACTIVITY . . . . . . . . . . . . . . . . . . . . . . . 37
5.0 SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . 40
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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1.0 INTRODUCTION
1.1 Statement of Work
1.1.1 Objectives
The :,:jectives of this contract were to identify and resolve
problems associated with the Orbiter/IUS communications systems.
1.1.2 Stipulated Tasks
The tasks associated with this contract include participation
in design reviews, coupled with acceptance of action items to be under-
taken for the resolution of review item dispositions (RID's). This in-
cludes reviewing all performance analyses submitted by the transponder
contractor.
2
2.0 GENERAL APPROACH
In the process of carrying out the required tasks, Axiomatix
has worked closely with the cognizant NASA personnel, the Orbiter prime
contractor (Rockwell International), the IUS prime contractor (Boeing
Aerospace Co.), and the IUS and Orbiter payload communication equipment
subcontractor (TRW Defense and Space Group). This activity includedI attending the Critical Design Review (CDR) and reviewing in detail all
communication-related performance analyses submitted by TRW.
While Axiomatix was engaged in these contractual activities,
TRW, the IUS STON/TDRS transponder contractor, received a stop-workI order since the Centaur is to be !.-:el as an upper stage for NASA missions,
and the NASA TDRS/GSTDN standard `.ansponder built. by Motorola was chosen
for use on the Centaur. Therefore, the remainder- of this report documented
the analyses and investigations undertaker, by Axiomatix and completed at
the time of the stop-work order.
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3
This section documents the reviews, investigations and analyses
of the TRW IUS STDN/TDRS transponder performed by Axiomatix. First, it
is appropriate to discuss where this equipment or subsystem fits into the
overall Orbiter/payload communication link. Figure 1 shows the top-level
IUS STDN/TORS transponder interfaces with the payload IUS and the Orbiter.
As can be seen in this figure, the transponder receives telemetry from
the IUS and passes commands to the IUS. The interface with the Orbiter is
via an S-band RF link with the Payload Interrogator (PI). Thus, the IUS f ,
STDN/TDRSS transponder must perform all the typical communication functionss
of acquisition, tracking, data demodulation and data modulation. These
functions will be addressed in subsequent sections.
Before analyzing these communication functions, however, it is
helpful to gain some perspective as to where and how these functions relate
to the overall transponder. This perspective is afforded by the transponder
block diagram shown in Figure 2. The specific detailed areas of the tran-
sponder involved in the Axiomatix investigation are indicated by the cross-
hatch lines in the transponder block diagram given in Figure 3.
r3.1 Analysi s of IUS STDN/TORS Trans ponder Performance
Volume I (Analysis) of the STDN/TDRS Transponder, S-Band Critical
Design Review (CDR) data package contains a series of detailed electrical
design analyses performed by TRW for Boeing that pertain to the manner in
which the STDN/TDRS transponder meets the performance specifications im-
posed by NASA for its use in IUS missions. Many of these analyses previ-
ously appeared in the Preliminary Design Review (PDR) data package deliv-
ered to Boeing on June 11, 1979. To the extent that these analyses (IUS
memos) were complete at that time, Axiomatix reviewed and critiqued their
contents and reported its findings shortly thereafter in Axiomatix Report
i No. R7911-5, November 30, 1979. Also included in that report were threei
appendices (C, E and F) which augmented and, in some instances, corrected i
several of the TRW analyses.
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Since these same IUS memos are contained in the CDR data
package, Axiomatix shall avoid duplication of effort by reviewing and
critiquing only ti.ose memos which were not contained in the PDR package.
In dealing with each of these analyses one by one, we shall simply refer
tn them by their IUS memo numbers, as per the entries in Table 1. It
should also be pointed out that most of the analyses were performed by a
single individual* (namely, Dr. Jack K. Holmes, Consultant to TRW) and
thus it should not be surprising that comments made on one particular
analysis might apply equally to many of the others. Such similarities
in approach, style, etc., will be indicated in our discussion so as to
avoid unnecessary redundancies.
l:
SThe remainder were performed by Dr. H. C. Osborne of TRW.
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Table 1. 'IUS Memos Reviewed and Analyzed
Section IUS Memo No. Title
2. TDRS Mode--General 125 TDI'SS Lock Detector Parameters,Structure and Performance forIUS
3. TORS--Carrier Acquisition 112 A Possible Problem in the I'ISAnalysis and Experiment Code-Aided Carrier Acquisition
Approach
118 Improved Estimate of the Code-Multiplied Spectral Density
114 Instantaneous Frequency Errorof the Carrier Loop Reference(at the Moment of Code VCXODisconnect)
116 Instantanenu; Frequency Error
of the Carrier Loop Referenceat the Moment of Code LoopDisconnect--Revisited
4. Costas Loop Performance-- 115 Revision r--IUS Phase DetectorAnalysis Biases Due to Hybrid and Arm
Filter Imperfections in theCostas Loop
117 IUS Slip Time
123 Influence of Arm Filter Delayon Tracking Performance of theIUS Costas Loop
5. SSP Analysis 110 Code Tracking Lock DetectorMean Time to Declare Out-of-Lock
6. STDN Dual Mode 122 STDN Dual Acquisition andTracking Analysis
124 Open-Loop Frequency Acquisition
7. STDN Only Mode 119 STDN Acquisition and TrackingLogic
120 STDN Acquisition and TrackingAnalysis
121 IUS STDN-Only Discriminator
Analysis for False Lock Avoidance
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3.1.1 IUS Memo No. 125
This memo discusses the design and analyses of carrier lock
detector performance for TDRSS dual-mode operation. In this mode, 2 kbps
(125 bps in the low rate mode) data is modulated on the 1-channel and
modulo-two added to the command spreading PN (Gold) code while the rang-
ing mode spread spectrum (truncated PN code) signal is modulated on the
Q-channel. The power ratio of the I- and 0-channels is 10 dB.
The lock detector, in conjunction with the AGC amplifier,
derives its error signal in the conventional manner from the I 2 -Q2 output
of a Costas loop. This error signal, which has the identical SxS*, SxN
and NxN components of the Costas loop tracking error signal, is filtered
by a narrowband (with respect to the arm filter bandwidth) lowpass filter
and the output is compared to a fixed threshold which is set at one-half
the threshold (minimum C/NO) signal level corresponding tr the low data
rate (125 bps) mode. The threshold outputs are used in a verification-
type algorithm to decide whether or not the loop is in lock. In particu-
lar, two successive below-threshold indications must occur to assure that
the lock detector declares the loop to be out of lock. If the loop is
indeed in lock when this occurs, a false dismissal then occurs. One
is interested in designing the mean false dismissal time to be quite long
(perhaps on the order of years).
On the other hand, when the loop is out of lock and a given low-
pass filter output sample exceeds the threshold, we have a false alarm .
Here one is interested in keeping the probability of such an occurrence
small so that the loop is not captured by frequent false alarms which
then require the verification algorithm to produce: two successive below-
threshold indications in order to finally declare the loop truly out of
lock.
In characterizing the performance of lock detection algorithms
- V as described above. the theory of finite Markov chains is particularly use-
ful. In this application, a three-state Oiagram is sufficient. with the
transition probabilities determin:d from ire lowpass filter output sta-
tistics. Since, as previously mentioned. the lowpass filter has a band-
iC width which is narrow compared to that of the Costas loop arm filters.
For this component, we must replace sin 24 by cos 2#.
t
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the filter output may then be assumed to hove Gaussian statistics, in
which case, the transition probabilities *,a• un the form of complementary
error functions of a threshold-to-noise ratio. The effective amount of
time one remains (wells) in each state is determined by the time constant
(r:orrelation time) of the lowpass filter (for state 1) and the specified
wait time after a below-threshold indication (for state 2).
Appendix I of this memo derives the mean and variance of the
false dismissal time (the time to reach the absorbing state (03) in the
Markov chain). For a small probability of missed cetection, the variance
is shown to be approximately equal to the square of the mean, which is the
same relation between the first two moments of an exponential distribution.
Assuming such a distribution for false dismissal time, the author easily
shows that the probability of falsely dismissing in t seconds or less is
1 - exp(, t/TFD), where TFD is the mean false dismissal time.Computations made from the above theoretical discussions reveal
a mean-time-to-false-dismissal of greater than seven years (for either
data rate) and a mean-time-to-dismiss a false alarm once the algorithm has
been put in the tracking mode of 175 m:. (for either data rate).
Based on the foregoing, the memo correctly concludes that
false alarms will not "capture" the system (since their probability of
occurrence is only 0.0016 and they are quickly dismissed) and false dis-
missals of true lock occur, on the average, infrequently enuugh (every
seven years) so as to "never" cause a problem.
3.1.2 IUS Memos Nc. 112 and 118
The TDRS carrier acquisition analysis and experiment section of
the CDR data package contains four* memos which can be arranged in two
pairs, nemely, 112 and 118, and 114 and 116. The second memo of each
pair represents a revision of the first and, as such, contains the more
meaningful results. Thus, we shall, in actuality. discuss only these sec-
ond memos (118 and 116) while, at the same time, pointing out the changes
made in the assumptions as given in the first memos (112 and 114).
IUS memo #112 calls attention to a possible problem (potential
malfunction of the code loop and carrier lock detector during code loop
loss of lock) in the lUS code-aided carrier acquisition approach. Thisf
Actually, there are five memos in this section, but one W07)was included in the PDR and, as such, was previously critiqued.
11
approach consists of locking the receiver VCXO to a multiple (192/31) of
the code loop VCXO with a CW loop, then multiplying this receiver VCXO
output signal up to S-band (a factor of 110.5) to serve as the code loop
input. The assumptions made in this memo are a 2-Hz single-sided code loop
noise bandwidth at threshold (C/No = 34 dB-Hz) and a carrier loop bandwidth
much wider than the code loop bandwidth. As such, the interaction (cas-
cading) of she carrier loop with the code loop was ignored and, thus, the
carrier loop was assumed to do no more than scale the code loop phase noise
process by the factor 192/31. Because of these oversimplifying assumptions,
the results of memo #112 led to the conclusion that the code loop would
potentially drop out in this mode and the carrier lock detector would not
detect the p: •esence of signal.
IUS memo #118 reconsidered the code-multiplied carrier acquisi-
tion problem under the assumptions of a 1-Hz (half as wide) code loop
threshold bandwidth and an identical carrier loop threshold bandwidth.
Certainly now the cascade of the code and carrier loop transfer functions
further reduces the spreading of the code clock multiplied carrier produced
by the multiplied-up phase noise of the code loop VCXO. Thus, despite the
fact that the line component of the carrier power spectral density is es-
sentially suppressed by the large phase noise obtained from the multiplied-
up code clock, all the spread component (although much wider than the orig-
inal phase noise process) is virtually contained within a ±100 Hz bandwidth.
In conclusion, then, the very wideband estimates of the code-multiplied
carrier power spectral density made in IUS memo #112 which cculd poten-
tially cause the code loop to drop out in this mode were, in IUS memo #118,
refined to the point where it may be safely concluded that the IUS code-
aided carrier acquisition technique is viable.
Axiomatix has carefully reviewed the analyses performed in these
two memos and agrees with the conclusions drawn therein.
3.1.3 IUS Memos No. 114 and 116
This second pair of memos in ^he iORS carrier acquisition anal-
ysis and experiment section addresses the problem of predicting the instan-
taneous frequency error in the carrier loop Just after disconnecting the
code loop VCXO when, prior to that time, the carrier loop was configured
to track a scaled version of the code loop clock. Indeed, an accurate
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estimate of this instantaneous frequency error is essential in deciding
whether or not the carrier loop bandwidth is sufficient to pull in this
frequency offset during acquisition.
Clearly, when the carrier loop is connected to the code loop
VCXO, an instantaneous frequency error of in the code loop would produce
a carrier frequency offset equal to (192/31)x(110.5)af = 684.4 af. Here
the first factor represents the scaling of the PN code clock at the
receiver VCXO input, and the second factor is the multiplication required
to bring this reference up to S-band. Thus, a one-sigma code loop fre-
quency error at threshold (C/No = 33 dB-Hz) on the order of 13 Hz (see
IUS memo #117) would produce, before disconnect, a carrier frequency off-
set of (684.4)(13)= 8897.2 Hz, which is so large that the carrier loop
could never acquire with any reliability. When the code loop VCXO is
disconnected, however, the action of the code loop and carrier loop fil-
ters reduce this frequency offset considerably, in particular, to a value
well within the frequency acquisition range of the carrier loop.
In IUS memo #114, the assumption is again (as in IUS memo #112)
that the carrier loop is wide compared to the code loop. Thus, the
approach taken in memo #112 was to compute (approximately) the standard
deviation of the voltage on the capacitor in the code loop filter, then
simply scale this quantity by the factor (192/31)(110.5)= 684.4 to arrive
at the carrier (one-sigma) frequency offset at S-band*. Multiplication of
this result by three (to give a three-sigma value) was then used to give
a rough estimate of the required carrier loop bandwidth during acquisition.
Interestingly enough, the computed value of carrier frequency
offset, namely, (192/31)(110.5/4)x(0.155 Hz)=26.5 Hz (0.155 Hz was the com-
puted one-sigma value of code loop filter capacitor voltage) was in excel-
lent agreement with a measured ,alue in the laboratory of 25 Hz. Unfor-
tunately, however, this was just a coincidence apparently caused by
t:
nullifying errors in the assumptions made in the analysis. In particular,
the variance of the capacitor voltage as given by equation (115) of this
memo is in rad/sec 2 , not Hz 2 , since the code loop VCXO gain, K v , is
Actually, the laboratory measurements were made at 25% of the
final S-band frequency; thus, the appropriate multiplication factor forcomparison of theory and experiment is (192/31)(110.5/4) = 171.1.
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in rad/sec/V. Thus, the calculated value of carrier frequency offset given
- previously, namely, 26.5 Hz, should be divided by 2,r, which results in
4.2 Hz. Furthermore, as discussed in IUS memo #116, the wideband carrier
loop assumption must be revoked in favor of a loop whose bandwidth is
identical to that of the PN code loop. When the two loops are now con-
sidered in cascade, the problem must be reformulated to directly compute
the RMS voltage (due to frequency offset) on the ca acitor in the carrier
loop filter. When this is done (as in memo #116), along with the 2v-fac-
tor correction previously discussed, then the one-sigma (RMS) frequency
error at S-band becomes (192/31)(110.5)x(0.01 Hz) = 6.84 Hz or, at the
laboratory measurement frequency, (192/31)(110.5/4)x(0.01 Hz) = 1.71 Hz.
These numbers correspond to threshold loop conditions, namely, both loop
dampings are at 0.707 and both loops have a bandwidth of 1 Hz.
Now since the theoretical values of frequency offset error are
considerably less than laboratory measurement values, the author points
out that this may be true because the analysis ignores the effects of
logic noise and oscillator noises. Indeed, since a large discrepancy
exists between measured and theoretical values, one might suggest that
these other unaccounted for effects tend to dominate. We hasten to add,
however, that estimates of oscillator noise and, in particular, logic
noise are difficult to come by, which makes accounting for these effects
analytically all the more difficult.
3.1.4 IUS Memo No. 115
This is the first memo in the Costas loop performance analysis
section of the CDR to be critiqued and discusses the static tracking phase
error induced by thermal noise biases at the output of the loop's third
multiplier. The two sources of bias discussed are an imperfect (other
than 90°) input hybrid and a combination of arm filter mismatch and input
bandpass filter asymmetry. Their effect, in producing static phase errors.
are treated independently.
In the hybrid misalignment analysis, the hybrid is modeled as
producing a pair of "quadrature" demoaulation reference signals which are
t• separated in phase by 90 0 + de, the quantity So representing the hybrid
angle error. When computing the phase detector outputs due to signal only
=t
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(as in eq. (10)), the author commits an error in that he ignores the hybrid
angle error. Thus, his final result for static phase error (eq. (14))
includes only the effect of hybrid angle error on the noise component at
the third multiplier output, which indeed turns out to be the less dominant
effect. To correct this error, we suggest that equations (10), (11), (12)
and (14) read as follows:
ec (t) d(t) cos4 ; es (t) _ 3P d(t) sin(m -60) (10)
e = aP sin(f-de)cosf
aPrsin -de _ sin de (11)LL
n2
P + 1 de = fss (12)
and
(Nef
ass -2aP + 1) de (14)
Then, for the case where R b = 2000 bps, f0 = 2000 Hz, a= 0.84, and P/N0 =
43 dB-Hz (includes IAB despreader loss), Table I of the memo, which tab-
ulates static phase error versus hybrid angle error, should be modified
as follows:
ae(°) oss(°)
1 1.187
2 2.375
3 3.56
4 4.75 0
5 5.94OF Pict (~,`.:.o i r
Thus, the statement made in the summary of the memo, namely, that "the
steady-state error is about 1/5 of the hybrid angle error" should be
changed to read "the steady-state error is slightly greater than the
hybrid angle error."1
a
15
The second part of this memo assumes a perfect (90°) input hybrid
and examines the effects of mismatched arm filters and an input bandpass
filter (BPF) with an asymmetric frequency response around its center fre-
quency. In particular, the two lowpass arm filters are assumed to be
one-pole RC filters with different 3-dB cutoff frequencies, namely, f l and
f2 , and the asymmetry in the equivalent lowpass version of the input BPF
is modeled as a linear "tilt" in the corresponding power spectral density,
namely, N
SL(f) = 0(1+af); jfj <5 kHz.
The author then shows that the static phase error produced by these two
sources of filter imperfection is given by
N
ass T' ( a0,P (f2-fI)f f
where many simplifications in the analysis were made by letting f' equal
fl or f2 in some of the manipulations.
For a 0.1 dB tilt at f= 5 kHz, (10 log l0 (l+ ax5x103 ) = 0.1, or
a= 4.88x10- 6), a= -0.5 dB, P/N 0 = 43 dB-Hz, fl = 1948.3 Hz and f2 = 2096.6 Hz,
the computed value of ass
(assuming f'= f 2) is 0.0038°. The memo follows
with a table which computes ass for larger values of tilts. The latter
three values of ass
in this table, corresponding to respective tilts of
0.3, 0.5 and 1.0 dB, should be corrected to read 0'.0108, 0.018 and 0.03920.
One further point of correction, although probably of second-order
importance, deserves mention at this time. The parameter a in the above
equation which ordinarily characterizes the arm filtering degradation on
the SxS term in the loop when both arm filters are identical should be
modified for the case where the arm filters are different. In particular,
we would now have (analogous to eq. (42)) for the noise effects
foo[Hl(w) H2*(w) + H1*(w) H2 (w) Sd(w) dw
16
where Sd (w) is the data modulation spectrum and H l (w), H2 (w) are the am
filter transfer functions, i.e.,
H i (w) _ —f- i=1,2
1 4 3 f.i
Substitution of H i (w) into the above expression for a and simplifying yields
(l+x'11+`^w l ! w2 ) dwa = ^2^ 2 Sd(w) 2n
1 + l wl^ 1 + \w2/
which, for a small 3-dB cutoff frequency difference, is approximately a,
as previously computed for identical arm filters.
Thus, in conclusion, the hybrid imperfection effect dominates
over the imperfect filtering effects, and the static phase error induced
is on the order of the hybrid angle error.
3.1.5 IUS Memo No. 117
The mean slip time of the carrier-tracking loop in the IUS-TDhS
transponder is computed at both threshold conditions CC/N0 = 33.7 dB-Hz,
R = 125 bps) and strong signal conditions CC/N 0 = 43.7 dB-Hz, R B = 2000 bps).
The loop is configured as a standard Costas loop with an input signal hav-
ing an unbalanced QPSK format characterized as follows. The received sig-
nal has a PN spread data modulation on the strong (I) channel and PN only
on the weak (Q) channel. The power ratio is fixed at 10:1. After being
despread by the I-channel PN code, the signal retains an unbalanced QPSK
format with data modulation only (assuming "perfect" despreading with a
fixed despreading loss) on the I-channel, and PN only (the product of the
in-phase and quadrature PN codes) on the Q-channel. This signal serves as
the Costas loop input. As such, the evaluation of the loop's phase error
variance due to thermal noise follows along the lines of previous analyses
of bi hase Costas loops with passive arm filters and unbalanced QPSK in-
puts. In making this statement, we tacitly make the assumption that the
17
PN code on the Q-channel behaves as a random data modulation of rate R c =
fc , where f is the PN chip rate (i.e., 3x106 Mchips/s). Thus, it is notsurprising that eq. (22) of the memo agrees with [1, eqs. (28) and (30)]
after the appropriate changes in notation.
Next, the memo evaluates the phase error variance component due
to oscillator phase noise. The phase noise model, based on IUS phase noise
specifications, was assumed to have a power spectral density which varied
as O f6 . For simplicity of computation of the phase error variance due tophase noise, the out-of-band loop transfer function 1- H(f) was assumed to
behave like a "brick wall" filter having zero value below the loop natural
frequency and unity value above this frequency. Finally, the two phase
error variance components ( that due to thermal noise and that due to oscil-
lator phase noise) are added to give the total phase error variance a202.
Before determining the mean slip time of the Costas loop, one
needs, in addition to the total phase error variance, the steady-state
phase error due to dynamics such as a residual carrier frequency rate of
of Hz/sec. For a second -order Costas loop, these two parameters are
related by
2nof
20ss 2 2(wn
where w is the loop's radian natural frequency which, for a 0.707 loop
damping, is related to the loop bandwidth BL by wn = 1.89 B
L. Having now
determined a202 and 0ss , the author computes mean slip time T (normalized
by the loop bandwidth) from the formula
BST = 1.5 exp 1 ' 2( 1 - sin 2fss,
a20
1 , 1.2 - sin _ 4,ref 1 2.5 exp 1 si --+
( )
1a2^
2(1.89^Bt2
(1)
28
This relation is valid for a second-order Costas loop with an active loop
filter and was originally obtained from simulation results on an analogous
phase-locked loop.
The author concludes with an evaluation of (2) for threshold and
strong signal conditions, and Ai = 70 Hz/sec. At threshold, a value of
BL = 25 Hz maximizes T, whose value is 6000 sec (10 min). In the absence
of phase noise and loop dynamics (Ai), T is monotonically decreasing with
increasing 8L. At strong signals, the same B
L = 25 Hz produces T>> 104 min.
The results given in this memo are obtained by straightforward
application of previously derived results and, as such, need no further
i. investigation.
3.1.6 IUS Memo No. 123
The effect of the delay induced by the arm filters in the IUS-
TDRS Costas loop on loop bandwidth and, hence, the phase error variance, is
investigated. The key step in the analysis is the approximation made with
respect to the signal e u (t) appearing at the upper arm filter output,
namely, that the effect of this filter on the data modulation and the loop
phase error are separable. More specifically, letting H(s) denote the arm
filter transfer function, then
eu (t) _ ►V H(p)[d(t) sin #(t)] (1)
which, for small 0. becomes
eu (t) _ F H ( p )[d(t) 4(t)] (2)
is approximated by
eu (t) _ A (H ( p ) d(t))(H( p ) 0(t)) (3)
where p has been used to denote the Heaviside operator. It is argued that
(3) follows from (2), provided that "the lowpass arm filter H(s) does not
seriously distort the baseband data stream." Although there appears to be
no approximate mathematics that can lead one from (2) to (3), there is a
t`
19
reasonable plausibility argument that one can use to make this step
somewhat believable. Typically, the #(t) process being slowly varying
with respect to d(t) appears as an envelope modulation on d(t) which,
when passed through the arm filter, is essentially unaffected in ampli-
tude but is shifted (delayed) by the arm filter group delay Thus, if
we approximate #(t) as a single-frequency (say, w 0 ) beat note, then
H(t) #(t) = #Ct- td, where t0 = arg Q wd/w0 is a good approximation
to the envelope modulation on the filtered data stream.
Making the above approximation, the author proceeds to find a
simple relation between the loop bandwidth (including the arm filter
delay effect), say B L (D), and the zero-delay loop bandwidth BLO , namely,
2
BLO1 - 2c amO
D 2
(4)
where C is the loop damping and D = w noT, with wnO
the zero-delay radian
natural frequency and T the time constant of the single-pole arm filter
H(s). Since wnO
and B LO are related by
NO = 2B LO( 4; ( 5 )1 +4C
then (4) can be alternately be written as
BL (D) - 1 0^)
28 L 1 1+ -f
ao0
Clearly, the mean-square phase jitter with delay becomes unbounded when
BLOT = (1 + 4c2)/4 .
(7)
t:
^ti
20
For c = 0.707, DLO
2 75 Hz and a 100-Hz 3-dB cutoff frequency (i.e., T =
1/2x(1001 = 0.00159), the increase in RMS phase jitter is only 9% cc4(0)/
a®01.09).
The author follows the computation of mean-square phase jitter
with a discussion of the effect of the arm filter delay on loo; stability,
as determined by Routh's stability criterion, and the root locus plot.
The interesting (but not too surprising) result is that the loop bandwidth
at which the loop becomes unstable is also determined from (7), namely,
the same value at which the mean-square phase error variance becomes
unbounded.
Finally, we wish to call attention to a similar study [2) with
similar results in which the effect of delay on the loop bandwidth and
stability of a data-aided loop (DAL) were investigated, thus lending more
credibility to the analysis performed in this memo. The DAL, which is
also used for tracking suppressed carrier signals, has much similarity to
the conventional Costas loop.
3.1.7 IUS Memo No. 110
This memo is the only one in the SSP analysis section which was
not previously critiqued by Axiomatix. In particular, it addresses the
mean time to declare out-of-lock for the code-tracking loop, both when
the signal is present and when it is absent. The lock detector algorithm
is of the "n-out-of-n" type wherein n (typically, 16) successi ve below-
threshold events are required to declare an out-of-lock condition. If an
above threshold even occurs anywhere along the way, the algorithm returns
the system to its initial state and resets the below-threshold count to
zero.
The mean time to out-of-lock performance of such a discrete
time lock detector algorithm is best determined by modeling the algo-
rithm as a 17-state Markov chain (the 17th state being the absorbing
state, namely, an out-of-lock declaration) and applying the well-known
theory for such chains to this particular case. Actually, for n 4 5, a
formula for this mean-time performance was determined by brute force
(direct) calculation in a previous memo by the author (see TRW IOC No.
SCTE-50-76-275/JKH). Thus, this memo serves to merely formalize the
W
21
i
validity of this result for all values of n. In particular, the mean
time to out-of-lock, T,•is simply given by
= nlT _ Tq DWELL
where TDWELL
is the dwell time per state (assumed equal for all states),
i.e., the time between threshold tests of the integrator output, and q is
the probability of a below-threshold event for any given threshold tEst.
Since, when signal is absent, q = 0.95 and, when signal is pres-
ent, q - 0.5, then for n = 16 and a 50-ms dwell time; the corresponding
values of T are found to be 1.27 seconds and 109.2 minutes, respectively.
The straightforward nature of these results and the absence of
complicating assumptions requires that no further investigation be
performed.
In the STDN dual mode section of the CDR package, two memos
were written which pertain to the analysis and design of the lock detector,
noncoherent, AGC and open-loop frequency acquisition circuits associated
with the carrier-tracking loop of the IUS transponder. Since the first of
these two memos (#122) assumes knowledge of the second (#124), we shall
start by critiquing the second.
3.1.8 IUS Memo No. 124
In the STDN dual mode of the IUS transponder, an open-loop fre-
quency acquisition scheme is used which involves linear sweeping of the
VCO frequency to bring the initial frequency uncertainty within the Pull-
in range of the loop (typically on the order of the loop bandwidth).
Since the loop is open during this sweep interval, an auxiliary detection
circuit must be used to determine when to remove the sweep and simultane-
ously* close the loop. This auxiliary detector consists of a coherent
amplitude detector (CAD) followed by a lowpass filter and threshold device.
In the actual frequency acquisition scheme used in the STDNmode, the sweep continues for an additional 4 ms after the detector indi-cates acquisition has been achieved to all for the processing time of themicroprocessor which controls the closing of the loop.
22
An instantaneous crossing* of the threshold by a signal at the input to
this device indicates acquisition whereupon the sweep is terminated and
the loop closed.
Such a half-wave rectifier type of open-loop frequency search
circuit has been previously described in [3]. This mew discusses its
application to the IUS transponder in the STDN dual mode. In particular,
computer simulation and laboratory test results are obtained for the wave-
forms at the output of the lowpass RC decision filter (in the absence of
noise) so as to enable selection of this filter's 3-dB cutoff frequency s
for a given sweep rate R (Hz/sec), normalized (to the peak signal ampli-
tude) threshold level d, and closed-loop bandwidth f l . Indeed, it is
shown that if, for a given initial frequency offset outside the loop's
pull-in range, B is too small, then, depending on the initial phase dif
ference 40 between the input signal and the swept VCO, the threshold may
or may not be exceeded as the VCO is swept through the pull-in range.
Increasing s helps this situation; however, if S is too large, then the
threshold is exceeded while the loop is still outside its pull-in range.
Hence, the sweep will be terminated and, consequently, the loop closed
prematurely.
The author provides what appears to be a reasonable rule of
thumb for the selection of s, namely, the peak value of the normalized
detector output frequency response H(f), evaluated at the edge of the
pull-in range (assumed equal to the loop bandwidth f l ) should be less
than the normalized threshold a. For a single-pole decision filter (B =
3-dB frequency), it is straightforward to show that the above is equiva-
lent to the conditiont
< fl s
Again. because of the 4-ms processing time of the micropro-cessor, a "stretching" or hold circuit follows the threshold detector toprevent situations where the input signal reverses and falls belowthreshold in less than 4 ms, i.e., sharp peaks.
'The author does not actually write this inequality in thisform although it is obtained by obvious steps from the results giventherein. Also note that this result is Independent of the sweep rate Ralthough the actual simulation results were performed for R-4000 3 Hz/s.
is
23
Thus, for d = 0.5 and f l = 400 Hz (STDN parameters), we obtain Q < 400/
d3 = 231 Hz (the author uses the approximate value 250 Hz).
The next area of investigation was the calculation of acquisi-
tion probabilities which were performed by computer simulation (in the
absence of noise) in view of the difficulty of obtaining these results
analytically. Note again that, although the additive noise was assumed
to he Absent, the probability of acquisition is, in general, less than
one u;ie to dependence of the acquisition process on the initial phase
difference +0 . The author compares the acquisition probability results
obtained by the above-mentioned simulation with experimental results
obtained in [3]. In some cases, there appears to be reasonable agreement
whereas, in other cases, there seems to be no match at all. Since, for
the latter situation, the author of [3] does not state to which of the
three possible open-loop implementations (one mixer and one half-wave
rectifier, one mixer and one full-wave rectifier, or two mixers and two
full-wave rectifiers) his results apply, one is unable to resolve the
discrepancy. Herein lies one of the principal rea jns for issuing IUS
Memo #124 in the first place, namely, to point out the lack of agreement
between the previously published experimental results and the computer
simulation results obtained by the author of the memo.
Finally, this memo concludes with a discussion of how the results
might be extended to account (in a very rough sense) for the effects of
additive noise.
In the opinion of Axiomatix, the results documented in this memo
represent a significant contribution to the understanding of the perfor-
mance and behavior of open-loop frequency acquisition techniques of the
type described therein. As such, the results are given in a sufficiently
general parametric form as to he useful in applications outside of the
IUS transponder. Perhaps the only area which would require further inves-
tigation would be the noise-present case, where computer simulation could
again be used (although with more difficulty) rather than the rough exten-
sion (valid only for high SNR) approach given in the memo. Indeed, the
entire subject of frequency acquisition in noise is an area of research
where much needs to be done.
1
i^
24
3.1.9 IUS Memo No. 122
Associated with the open-loop frequency acquisition technique
described in IUS Memo #124 is the lock detector of the STON dual made
whose functions are to close the tracking loop and stop and sweep* when
frequency acquisition has been completed. The indication that frequency
acquisition is complete is a high (a°)ove threshold) signal from the
sampled-and-held output of the threshold device in the frequency acquisi-
tion circuit. Thus, this signal serves as the input to the lock detector
whose control algorithm is as follows: When the loop is initially open,
a single high sampled-threshold output shall close the loop. Two succes-
sive high-threshold „dtputs are required to terminate the sweep. Also,
when the loop is initially closed and in lock, two successive low-threshold
outputs are required to open the loop and reinstate the sweep.
The purpose of this memo is to determine decision filter band-
width and threshold settings for the above circuit, taking into account
both the hold circuit at the sampled-threshold output and the noncoherent
AGC (NAGC) which accompanies the loop. As in the previous lock detector
analyses (see IUS Memos #'125 and 110), the theory of finite Markov chains
is used to determine the mean time to false alarm (falsely close the loop
and falsely disable the sweep) performance. Other computations include
probability of false alarm (falsely closing the loop and falsely disabling
the sweep) and accidental restart (falsely open the loop).
For the NAGC effects, the author assesses the increase in AGC
gain in going from acquisition CS/N0 = 46 dB-Hz) to tracking Ceffec:ivetS/N0 = 40 dB-Hz) as a function of the AGC filter bandwidth. Also deter-
mined is the further increase in gain (up to a practical limit) when only
noise is present. The analyses performed here is similar to that done in
IUS Memo #125 and, as such, requires no further explanation. Similarly,
the false alarm and accidental restart probability ,--Iculations parallel
those performed in IUS Memo #125 (except for the effect of holding the
threshold output sample for 4 ms, which is shown to roughly double the
false alarm probabilities which would be calculated for an instantaneous
Actually, the lock detector telemeters a message to the groundand the ground stops the sweep.
tThe actual S/NO during tracking is 43 dB-Hz; however, the addi-tional suppression caused by the presence of command modulation and, pos-sibly, two ranging tones, both phase-modulated on the carrier, is 3 d8.
25
sampling operation). Since the loop is closed after the first threshold
crossing and the sweep disabled after the second threshold crossing, the
author computes the "instantaneous" probability of falsely disabling the
sweep as the square of the probability of falsely closing the loop. This
is only approximately correct since the probability of exceeding the
threshold the second time must be computed with the loop closed, while the
probability of exceeding the threshold the first time is computed with the
loop open. In general, these two threshold crossing probabilities will be
different, depending on how far out of lock (amount of frequency offset
relative to the loop's pull-in range) the loop is by the time of the second
threshold crossing.
Analogous to the difference in the false alarm probabilities
for closing the loop and disabling the sweep, the mean time to occurrence
of these false alarm events must be computed from different Markov state
models. For the former, the mean time is simply
_ TO
Tclose p
where TO is the threshold sampling time interval (i.e., 4 ms) and p is
the threshold crossing probability when the loop is out of lock. For the
latter, the mean time is*
_ TO + (1 - q) T1
Tdisable(1-q)
where T1 is the time the loop remains closed after the initial closure
before the threshold is again sampled, and i- q is the probability of
exceeding the threshold when the loop is in lock. Here again, the above
result is only approximately correct since it assumes that the transition
probabilities from state 1 (loop closed) to state 2 (sweep disabled) are
the same as those from state 0 (loop open) to state 1.
*The author's result for this quantity, namely,
TO (1-q)T1_Tdisable
(1
is incorrect although the numerical evaluation appears to be correct.
26
To compute the probability of accidental reopening of the loop,
the author points out that three cases can occur. For the first case, the
assumption is that the loop has locked but the sweep is still on; hence,
the loop is tracking the sweep with a steady-state phase error equal to
the arc sine of the ratio of the sweep rate to the square of the loop's
natural frequency. For the second case, the loop is tracking, but the
NAGC has not yet had time to act. Finally, the third case is the same
as the second except that the NAGC has now had time to act. This last
case yields the largest restart (reopening of the loop) probability and,
hence, represents the worst case.
The memo concludes with the corresponding mean-time-to-loss-of-
lock calculations which employ a Markov state model analogous to that
used in computing mean time to disable the sweep, the difference being
that the above-threshold probabilities are switched with below-threshold
probabilities and the latter computed assuming an in-lock condition, i.e.,
tracking.
In summary, the computations are straightforward applications
of the Markov chain-approach, the theory of which was documented in pre-
vious IUS memos. Thus, other than the modifications to include such
effects as sample-and-hold time, the results are analogous to those pre-
viously obtained for the TDRS carrier and PN code loops.
3.1.10 IUS Memos No. 119 and 120
These two memos are for the STDN-only mode, the companions to
IUS memos 124 and 122 for the STDN dual made of operation. Again, the
purpose of the documentation is to characterize the behavior and analyze
the performance of the acquisition and lock detector schemes, along with
a determination of the necessary threshold settings. Although, in prin-
ciple, three possible frequency acquisition schemes are under considera-
tion, namely, (1) an open-loop scheme similar to that for the STDN dual
mode, (2) an existing digital hardware version of a closed-loop scheme
and (3) a software implementation of (2), only the second scheme (also
referred to as the DOD version) is discussed in these memos. Since this
scheme is referred to as a "closed" loop frequency acquisition technique,
it implies that the VCO is swept with the loop closed and, thus, a high
F
27
(greater than threshold) signal out of the lock detector is used only to
stop the sweep. In reality, the VCO is initially swept open-loop and can
be immediately (with no delay) closed by a low (less than threshold) out-
put from an auxiliary discriminator circuit prior to a lock detector
threshold crossing, or subsequently (after a short delay) by a lock detec-
tor (high) signal* itself, with the former being the more likely to occur.
Once the loop is closed, however, only the lock detector output signal
both stops the sweep and maintains the closed loop after the sweep has
stopped. It is in this sense that the behavior of the lock detector is
analogous to that of the STDN dual mode of operation.
The behavior of the actual closed-loop frequency-searching cir-
cuit employed closely parallels that previously described in [3], the main
difference being that, during the initial part of the sweep, the loop is
open until closed by the discriminator. In addition, after the loop is
closed and the sweep has been stopped, the loop bandwidth is narrowed for
the tracking mode of operation.
Assuming that the loop has locked and reached the steady state
(the discriminator has previously indicated that the loop be closed), but
the sweep has not yet been removed. The DC output of the coherent ampli-
tude detector (CAD) is then simply given by
u0 = VJ1 JO(1.1)y f 1 - (R/wn') 2 (1)
where S 1 is the signal power at this point, R /2w = 106 Hz/sec is the
sweep rate, and w n = 1.89 BL = 3780 rad/sec is the natural frequency of the
loop, with BL = 2000 Hz the loop noise bandwidth. The J 0 (1.1) factor
occurs because the input carrier is phase modulated by a data-modulated
16 kHz sinusoidal subcarrier with modulation index 1.1 radians. Thus,
once S 1 is determined (depending on the action of the noncoherent AGC),
u0 is specified. Furthermore, the variance of the noise at the CAD filter
output also depends on the NAGC action, i.e., whether or not signal is
Actually, two possibilities exist here, namely, a single highpulse of greater than 3-ms duration, or two or more short (less than 3-msduration) pulses within 33 ins (but greater than 3 ms apart) will closethe loop after a total delay of 36 ms after the leading edge of the firstpulse.
28
present. The AGC gain is normalized such that it has value unity when
signal is present and S%N O = 52 dB-11z. Thus, when signal is present, the
CAD decision filter output has a DC value given by (1) with S 1 = S and a
noise variance vu l 2 = NOB, where B is the noise bandwidth of this filter
(a single-pole RC filter). When signal is absent, the DC output is zero
and the noise variance is
(112 = C1 + N S
) N
OB (2)0 AGC
where BAGC is the noise bandwidth of the AGC input filter [also, a single-
pole RC with 3-dB cutoff frequency of 1600 Hz, i.e., BAGC = (n/2)(1600) .
Using these relations and some additional results given in [3] for prob-
ability of successful acquisition, it is a relatively straightforward
matter to compute the false alarm and detection probabilities associated
with the action of the lock detector in stopping the sweep. One further
assumption is made that the threshold d is chosen equal to half the DC
value of the decision filter output corresponding to signal present, i.e.,
d = 1/2 uO , where u is given by (1). This yields (for a decision filter
bandwidth of 200 Hz) a false-alarm probability* of 10-3.
3.1.11 IUS Memo No. 121
(a) Introduction
The purpose of this memo is to give an approximate analysis of
a discriminator used to avoid false locks and to specify the filter band-
widths and number of poles needed for these filters. The two latter tasks
seem to be rather straightforward, and no major objections arise, at least
insofar as the resulting SNR is the dominant measure of performance for the
last case.
The analysis of the discriminator is basically composed of two
similar parts: Case A/B and Case C. The first part (A/B) tries to esti-
mate the probability that, as it sweeps through the subcarrier component
*Actually, this false alarm probability is the probability thattwo or more samples of the CAD decision filter output exceed the threshold6 in a 65-ms interval.
29
of the spectrum (as shown in Figure 4), the discriminator will indicate
a voltage below threshold which would imply a false-lock case. In other
words, the phase-lock loop would track that component, while the discrim-
inator would falsely indicate (by being below threshold) that this is
actually the carrier component. Ideally, this would not occur if noise
were not present because the discriminator window centered at the first
subcarrier harmonic would "see" an asymmetric spectrum, consisting mainly
of the carrier component at the edge, and would thus produce an output
above the threshold. Note that the distance between the carrier and the
first subcarrier is 16 kHz, while the discriminator window is 25-kHz wide
(one-sided). Hence, the carrier is well within its reach when the window's
center is at fcarrier + fsubcarrier. Therefore, it is the noise which
might cause a false indication. Hence, the analysis of section C has a
similar objective and it addresses the case where the discriminator is
centered at the carrier which should, ideally, provide an indication below
threshold. The analysis then aims at the probability of false rejection.
Since the analytical tools are identical in both cases, we examine in
detail only the first. It should be stated that the overall system is
highly complicated (nonlinear/time-varying/stochastic), and thus, it is
possible that both the original analysis and our critique can be subjected
to further questions regarding the accuracy of some of the approximations
and analytical techniques. In all fairness, however, we should recognize
that the analysis is very close to the limits of analytical techniques,
exempting, of course, any numerical mistakes and, possibly, some minor
theoretical improvements. We note at the very onset of the analysis how-
ever, that there is a fundamental question concerning the validity of the
approach. The question is: since the system is time varying (because of
the sweeping process), can steady-state analysis provide credible results,
especially when calculating noise variances? Assuming that the above is
answered positively (otherwise, the whole analysis collapses), we will
proceed with side observations/corrections/improvements, as listed below,
further indicating which ones might bear significance to the final results.
' 30
Case A
i
kHz kHzCase B
iii
Knz Knz
Case C
kHz kHz
C r. r'r'r.Y
Figure 4. The Three Cases of Discriminator Detection of Interest
31
(b) Specific Comments
Since the carrier-tracking loop is a phase-lock loop, it should
not lock to the subcarrier if the subcarrier is biphase modulated by com-
pletely random data. This modulation, of course, results in a (sin x/x)2
spectrum. If there is a periodic component or long string of all zeros,
or all ones, however, in this data, then the carrier loop can lock. Thus,
the analysis performed by the author is a worst-case analysis since it
assumes no data modulation on the subcarrier.
The author's expression for sweep rate as determined by Viterbi
(7) has an extra factor of 106 in it and should be stated as
W 2
R N F = - = 1.14 MHz/sec.
Likewise, the sweep/rate for the STDN-only mode (8) has an extra factor of
106 and should be stated as
R = 1 MHz/sec.
Also, the expression for the outputs of the in-phase and quadrature mul-
tiplexers, (19) and (20), erroneously have w 1 rather than w0 , and the
expressions for the arm filter outputs of these signals, e c (t) and es(t)
((24) and (28)), should have n l (t) and n"2 (t) for the noise terms rather
than n i (t) and n 2 (t). We also note that, by introducing a ew parameter
(ew = frequency from fcarrier + fsubcarrier) and proceeding with the anal-
ysis, one assumes that ew is fixed, a contradiction to the very fact of
the sweeping mechanism. Pursuing this, we see that the factor Hi2(ow)
has been omitted from the expression for the input SNR, p i , for the lim-
iter (48). Correctly stated, p i is given by
P J 1 2 (1.1) H12(ew)
Pt 2 N 0 B 1 -
The significance of that is rather minor if nw is assumed to be well
within the flat portion of the filter. However, ew is variable with
time. Furthermore, in the expression for p i , the author has used J12(1.1)
incorrectly, instead of J O 0.1). This means that the numerical evalua-
tion of p i should be corrected accordingly. Thus, p i should be -3.48 dB
W432
V(not 0.24 dB), or a factor of 0.4468 (not 1.06). Since the SNR pi is
used to evaluate the noise variance, the error might be important in the
sense that the noise is underestimated.
The noise spectra calculated by the author and shown in Figure 5
are confusing. The sum bound comes from a calculation which is not shown
and is of questionable value since it is not used anywhere in the follow-
ing text. Furthermore, the values of AL (hard-limiter voltage gain) and
ow are not indicated anywhere. Spectrum SN2 (f) has been plotted for some
value of em, but the sum of SN 1 (f) + SN2 (f) does not match the sum spectrum.
The purpose of the overall argument is to substitute (or upper bound) the
sum of noises
F AL -p i/2 2F AL -p i/2 (pi
e IO (pi)
2 n l (t) + e Ill T) Yt) sin(2Awt+28)
F a F a
with a single noise process n l (t) with an appropriate gain. The simplest
way to do it would be to neglect sin(-) since sin( . )) < 1, then upper
bound the sum as
Pi^
3f AL -p i /2 pi e I 0 2( )n 1
(t) 1+a I
(P P)
Then, for the range of p i of interest, substitute the ratio (Ii(-))/IO(•)
with an appropriate number of the order of 0.2 -0.5. Instead, the analy-
sis uses the factor 3 -.7 without further justification. Also, the author
states that the resulting noise is essentially flat up to 30 kHz. That
is obviously a pessimistic assumption since n l (t) is simple white band-
pass noise filtered by LPF 1 , which is a one-pole filter with cut-off fre-
quency of 25 kHz. However, this is not a bad assumption from the point
of view that it upper bounds the noise contribution and simplifies the
following analysis.
We should point out the following: In the numerical results,
it seems that the author uses the value of = 0 since he is interested in
the case where the discriminator is centered exactly at f + f sc,
If that
33
0N
O
b 4-S-41 NU ZO NCLN rd t
E LO +j —
N U 4- M
CL CY)
O N Z J4- E NC IC N II
N .^O ^.00N ^
I
^ Z
II I Lnr.
4-
r-1f
L
UaIZN410.r
OZ
COm
1LOIZd
N
CS bY
GJ
OaEOV
b7•r
^r
C
L(
L
Q1
LLW LO Ct M N ^--1
-+ 0 0 0 6 8 O O 8 O
34
is indeed the case (and there is no contrary evidence since the value of
of does not appear anywhere), all the calculations for Se 3 (0) and Se2(0)
are useless becuase they will be zero for of - 0 ((68) and (70)).
In a very heuristic coupling of transient to steady-state con-
cepts, the author models a time-varying input to OF 31 as shown in Fig-
ure 6. The signal level at the output of LPF 3 is calculated based on that
input and, from that, the resulting probabilities are calculated. There-
fore, such an assumption really affects the overall conclusions of the
analysis. This statement is made in light of our lack of faith in using
steady-state analysis techniques to calculate the performance of essen-
tially time -varying phenomena. However, an exact assessment of the ap-
propriateness of this assumption is beyond the scope of this review.
For the case of only two signals and noise into a hard limiter,
as is the case here, the results of Jones [4] will yield a more accurate
answer than Shaft's [5] expression used by the author.
Thus, using Jones' results, we have the carrier and subcarrier
components, respectively given by
S11/2 W (-1)i(7) ((• 1 52
b010 = n (LNO r 1i +7) 2F1 -i,-i,-1+1+ S
i =0 i10+011
and
^1 $2
S1^2' (-1)iC N /
11 S2
b001 n \ N / ^ 2 r(l +i
F1 -i;-i;2,=0 (i!) 1
where
A2 A2S 1 =——; S2 = 2- ; N= a2.
We also note that the author's expression for the discriminator
dynamic DC output voltage ((95))
8 a 2 A 2(75)S = c- 2 z(2w) ^16x 103
n
36
is calculated for 4w - 0; in which case, the second term in his expression
for the noise variance
PPo2 - 3.9 AL f03 NO2 a-2p I
04 (2) 2Btp
♦ 76.8 NO a A L 4 2
T e-p IO2(2)
no
x (nf)2jH2(of)12 26
1 + /Af
LP
0/
should be zero. Finally, the rationale for the selection of the threshold
as Th -112 S is not provided.
In summary, we have found some minor flaws in the author's anal-
ysis; we also question the appropriateness of using steady-state analytical
techniques to analyze time-varying phenomena. However, we do not believe
that the minor flaws significantly change the conclu ,aions reached by the
author, nor do we have any alternate analytical techniques to suggest.
This type of system is difficult to analyze, and the author's efforts
probably represent the best that can be done in a reasonable amount of
time and at a reasonable cost.
37
4.0 CDR ACTIVITY
As part of its contractual activity, Axiomatix attended the
STDN/TDRS CD1. There were 21 action items and 26 RID's. The action items
and RID's primarily concerned the lack of test data to establish the com-
munication performance of the transponder over the link. Tables 2 and 3
present descriptions of the key action items and RID's, respectively.
It should be noted that the action items and RID's correspond in most
cases to the areas of analysis that Axiomatix is engaged in and documented
in Section 3 of this report. Action items that addressed new areas were
AI-5, AI-9, AI-10 and AI-21. These new areas were scheduled into the
Axiomatix analysis effort but, due to cancellation of the STDN/TDRS tran-
sponder effort, were not completed.
38
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4) NL NN U 4-3 "0 . t0 b+J Oi +) 3 +J 3 C m 4) N C O N = S-o d 0-0 t0 L L r r M Ln •r 4.) N O a(a L t f0 N i 4) O i ;:L 4) C i L () 4J•r r- N 0' 4) N t0 4) +J 3Y L O7 (U +J +0 7 O 3 4) •0 LO •r = O +) N 4J M7 +- +J +J 0'4-) O L. C • O +J Ln +-) O Ccc ^ :3 N f0 i t0 4) :C N O M C e-1 r N C 4J •rE.0 n z4) -0i 3 (A4) = 0. N N E to f01
L.^G " f ++ +J a ^ 4) rO- 0 4J C O CA 4N) b C 4J 4J UN 4) N b N L •r O t0 Ln ON t0 r• = f0 +•I OC. i-) N 4) Si 4) +J +1 N 10 O L.N • L +- i 4) L N rd (0 +J +-) t0 >> 4) ^4 41 r+ a Q) +1 E 4) 4) Ui
+' O f L 4) r- +► f0 4)N 4) O 4) O N t0 U 4) N NL Z C = .r L L L C X 4- L i 0 L O 4) C () 0) +J LF- a. O )-- N )-- +1 (0 Q 4) O F- 4- Ln F- +-) r- 1- +J L N +)
VU 4CiC •r
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OU3 4-)
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+1 O +t i O rf0 •r O O r OO N a i U Z Oi 4) 7 W f0 41 dO i O +-) N >4J a 4) t0 f6 (0t0 a i N O t 3r 4) f0 d
N +► L >> J0 I 4J 4. S- i 1QJ E U 4) > Nr N •r E r (o]L i C +1 4) 4J L
GN. af0 i ^ 4) NO' t..) F- V) F- tY co
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39
3•
40
5.0 SUMMARY AND CONCLUSIONS
The analyses,'reviews and investigations conducted by Axiomatix
prior to cancellation of the TRW IUS STDN/TDRS transponder effort uncov-
ered no basic reasons to believe that the transponder would not work sat-
isfactorily with the Orbiter. However, Axiomatix did uncover several minor
flaws in some of the analyses performed by TRW. It is doubtful that these
flaws would significantly change the conclusion stated above. Had the
effort continued, however, Axiomatix would have refined or reanalyzed the
areas in question. Furthermore, where some of the analyses for performance
in noise were very approximate by necessity, Axiomatix would have conduct-
ed computer simulations to verify parameter values and performance.
As a result of its participation in the IUS STDN/TDRS transpon-
der CDR, Axiomatix undertook several new analyses. Due to cancellation
or the transponder effort, however, only several of these were completed.
These are documented in Section 3 of this final report.
• 41
REFERENCES
1. M. K. Simon and W. K. Alem, "Tracking Performance of Unbalanced QPSKDemodulators: Part I--Biphase Costas Loop with Passive Arm Filters,"IEEE Transactions on Communications, Vol. COM-26, No. 8, August 1978,
PP -
2. M. K. Simon and J. C. Springett, "The Theory, Design and Operationof the Suppressed Carrier Data-Aided Tracking Receiver," JPL Tech-nical Report #32-1583, June 15, 1973.
3. A. Blanchard, Phase-Locked Loops: Application to Coherent ReceiverDesign, J. Wiliy—and Sons, 1976, Chapter 11.
4. P. Shaft, "Limiting of Several Signals and Its Effect on CommunicationSystem Performance," IEEE Transactions on Communications Technology,Vol. COM-13, No. 4, December 1965.